We review the class of inverse probability weighting (IPW) approaches for

We review the class of inverse probability weighting (IPW) approaches for the analysis of missing data under various missing data patterns and mechanisms. each subject BIIB-024 and the = (denotes the variables that are subject to missingness. We let R= (denote the vector of missing indicators for subject where the (1 ≤ ≤ is observed and 0 otherwise. Let V(Rdenote the observed components of Vdenote the observed data for subject and let L= V(1?Rdenote the unobserved components of V| W; β]. In missing data models with missing outcome and covariates W would represent the covariates that are always observed and V would include both the outcome of interest and the covariates that are subject to missingness. Throughout we assume that (W= 1 … are independent and identically distributed random vectors. We assume the parameter of interest β* is the unique solution to the equation = is observed for subject or it is completely missing. This pattern BIIB-024 often occurs BIIB-024 when information is extracted from multiple data sources. For example administrative claims data contain information on basic demographics (age gender) healthcare utilizations and medication dispensing records. However more detailed clinical information such as vital signs and lab test results would be available only for a subset of the study participants with linked EHR data. = indicates whether heart disease occurred during the 1-year follow-up period after drug initiation. Let ? = 0) of Vis missing then all subsequent elements are missing (= 0 for any < ≤ might denote the data that were to be collected at the records the BP at the end of a 12-month follow-up period. The baseline BP missing. We decide to make the data “monotone” by ignoring the data on | W =1 but = 0 for some subjects and = 0 but =1 for others. This is the most complicated missing data pattern. We consider two BIIB-024 motivating examples for this pattern. =1 2 3 As before W contains the treatment indicator and baseline covariates (e.g. age sex). Let indicate the BP measured at the = (= r | W= r | L= r | W=1 | W= 1] ≠ 0 then if we use complete cases only and estimate β* by solving the estimating equation = 1 | W=1 | Wor 0 = (0 0 …0 …0)with complete data (R= 1) by the inverse of the conditional probability of observing the full data = 1)is a consistent estimator of β* under regularity conditions.21 Moreover the asymptotic variance of where = 1)= 1)= 1)= 1)of Λ1 ≡ {(= 1)∈ = 1)equals = ? = 1 W] ? only since V(0) is an empty set. Thus since = r | W) = = r | Won the always observed covariates Wvia either a parametric regression model (e.g. logistic regression) or nonparametric data-adaptive algorithms (e.g. tree-based methods).31–35 In many studies that obtain BIIB-024 data from electronic medical databases the number of covariates that need to be adjusted for to make the MAR assumption plausible is quite large.36 Then it will be difficult to impose a correct parametric model for = 1 | W=1 … depends on the unknown outcome regression function = | R = 1 W] ? | R = 1 W]. As before we can use either a parametric working model | R = 1 W; is obtained by solving the augmented estimating equation is doubly robust (DR) in the sense that it is consistent for β* if either the working model for the missing data process | R = 1 W] is correctly specified but not necessarily both.38 This nice property offers analysts two chances of making correct inference. Furthermore the specified working models are practically certain to be incorrect especially in the presence of high-dimensional covariates. But as long as at least one model is nearly correct the bias of will be small by theory and simulation results.38 The variance estimates of can be obtained using either the asymptotic theory and delta methods or bootstrap re-sampling approaches. 4.2 MNAR The MAR assumption cannot be empirically tested using observed data except under limited scenarios.39 Subject matter expertise is usually required to judge its plausibility. When MAR does not appear to be reasonable then TIAM1 additional assumptions on the missing data process need to be imposed to make the parameters of interest identifiable. Since these additional assumptions are not verifiable under a nonparametric full data model for (W V) a sensitivity analysis is recommended. There are different ways of conducting a sensitivity analysis for MNAR (i.e. nonignorable) data. We focus on the selection bias function approach for IPW estimators.27 30 This approach decomposes.